CSSU Math Frameworks
The CSSU Math Frameworks is based on the National Council for Teachers of Mathematics (NCTM) Standards
and is aligned with the Vermont Expectations for Mathematics (Vermont Institutes.)
What are the NCTM Standards? Descriptions of the mathematical understanding, knowledge, and skills that
students should acquire from pre-kindergarten through grade 12
♦ Content Standards explicitly describe the content students should learn.
♦ Process Standards highlight ways of acquiring and using content knowledge.
Goals for students:
ƒ learn to value mathematics;
ƒ become confident in their ability to do mathematics;
ƒ learn to communicate mathematics; and
ƒ learn to reason mathematically.
CURRICULUM
INSTRUCTION
ƒ Instructional elements:
NCTM Principles and
Teaching Standards
ƒ Instructional Programs:
(Everyday Math, Connected
Math, Algebra courses,
etc.)
NCTM Content & Process Standards
ƒ Number & Operations
ƒ Algebra
ƒ Geometry
ƒ Measurement
ƒ Data Analysis & Probability
ƒ Process Standards (Problem Solving,
Reasoning & Communication)
ASSESSMENT
ƒ Vermont Grade Level Expectations for Math
ƒ State assessments
ƒ Local assessments
CSSU Math Curriculum Committee
CCS - Ginger Johnson, Gail Blasius
HCS - Eunice Branch, Kerri Wallis
SCS - Joan Cavallo, Joey Adams, Walter Nardelli
WSD - Kathy Shaw, Dave Lyons
CVU - Alison Sherwin, Monica Carter, Val Gardner
CSSU - Christine Hopkinson, Cris Toomey, Nancy Pollack, Pam Cyr
Table of Contents
NCTM Principles and Teaching Standards
page 3
Everyday Math and Connected Math
page 5
What is Computational Fluency
page 6
Content Standards
Number & Operations (N)
Algebra (A)
Geometry (G)
Measurement (M)
Data Analysis & Probability (D)
page 7
page 10
page 13
page 15
page 17
Process Standards
Problem Solving (P)
Reasoning (RES)
Communication (COM)
Connections (CON)
Representation (REP)
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NCTM PRINCIPLES
EQUITY: Excellence in mathematics education
requires equity and high expectations and strong
support for all students.
♦ High expectations and worthwhile opportunities for
all
♦ Accommodating differences to help everyone learn
mathematics
♦ Resources and support for all classrooms and all
students
Equity does not mean that every student should receive
identical instruction; instead, it demands that reasonable and
appropriate accommodations be made as needed to promote
access and attainment for all students.
CURRICULUM: A curriculum is more than a
collection of activities: it must be coherent, focused
on important mathematics, and well articulated
across the grades.
♦ Seeing how ideas build and connect with other ideas
♦ Mathematics that will prepare students for
continued study and for solving problems in a
variety of school, home, and work settings
♦ Opportunities to learn increasingly more
sophisticated mathematical ideas as they progress
through the grades
The strands are highly interconnected. A coherent curriculum
effectively organizes and integrates concepts so that students
can see how the ideas connect… enabling them to develop new
understandings.
TEACHING: Effective mathematics teaching
requires understanding what students know and
need to learn and then challenging and supporting
them to learn it well.
♦ Knowing and understanding mathematics, students
as learners, and pedagogical strategies
♦ A challenging and supporting classroom learning
environment
♦ Opportunities to reflect on and refine instructional
practices
To be effective, teachers must know and understand deeply the
mathematics they are teaching and be able to draw on that
knowledge with flexibility in their teaching tasks.
LEARNING: Students must learn mathematics
with understanding, actively building new
knowledge from experience and prior knowledge.
♦ The vision of Principles and Standards is based on
students learning mathematics with understanding.
♦ Conceptual understanding allows students to use
their knowledge flexibly and solve new problems.
♦ When students learn mathematics, they combine
factual knowledge, procedural facility, and
conceptual understanding in powerful ways.
Learning with understanding is essential to enable students to
solve the new kinds of problems they will eventually face in the
future. Effective learners reflect on their thinking and learn
from their mistakes. Classroom discourse and social
interaction can be used to promote the recognition of
connections...
ASSESSMENT: Assessment should support the
learning of important mathematics and furnish
useful information to both teachers and students.
♦ Assessment should be an on-going process in which
teachers measure student understanding in order to
guide their instruction and enhance student learning.
♦ Three types of assessment are pre-assessment,
formative assessment, and summative assessment.
♦ Assembling evidence from a variety of sources such
as open-ended questions, constructed-response
tasks, selected-response items, performance tasks,
observations, conversations, journals, and portfolios.
Assessment is an integral part of instruction that informs and
guides teachers as they make instructional decisions.
Assessment should become a routine part of the ongoing
classroom rather than an interruption… assembling evidence
from a variety of sources is more likely to yield an accurate
picture.
TECHNOLOGY: Technology is essential in
teaching and learning mathematics; it influences
the mathematics that is taught and enhances
students’ learning.
♦ Students can learn more mathematics more deeply
with the appropriate and responsible use of
technology.
♦ Technology cannot replace the mathematics teacher,
nor can it be used as a replacement for basic
understandings and intuitions.
♦ When technology tools are available, students can
focus on decision making, reflection, reasoning, and
problem solving.
Technology supports effective teaching, but should not be used
as a replacement for basic understandings and intuitions.
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TEACHING STANDARDS
Choosing an excellent mathematics series that meets the goals of NCTM Standards can be the first step toward revitalizing our
mathematics classes. But materials alone do not make the program. Another key to success is the teacher’s willingness to increase the
effectiveness of the materials by spending considerable time planning the lesson, listen carefully to what the students are saying in the
classroom, analyzing what they are learning, and consequently adjusting the mathematical tasks and the questions asked. In other words,
teaching matters. --Glenda Lappan, President, NCTM
WORTHWHILE MATHEMATICAL TASKS:
The teacher of mathematics should pose tasks
that are based on-
• Knowledge of the range of ways that diverse
students learn mathematics
• Develop students’ mathematical understandings
and skills
• Stimulate students to make connections
• Call for problem formulation, problem solving,
and mathematical reasoning
• Promote communications about mathematics
• Represent mathematics as an ongoing human
activity
TEACHER’S ROLE IN DISCOURSE: The
teacher of mathematics should orchestrate
discourse by-
• Posing questions and tasks that elicit, engage, and
challenge each student’s thinking
• Listening carefully to student’s ideas
• Asking students to clarify and justify their ideas
orally and in writing
STUDENTS’ ROLE IN DISCOURSE: The
teacher of mathematics should promote discourse
in which students-
• Listen to, respond to, and question the teacher
and one another
• Use a variety of tools to reason, make
connections, solve problems, and communicate
• Try to convince themselves and one another of
the validity of particular representations, solutions,
conjectures, and answers
• Rely on mathematical evidence to determine
validity
TOOLS FOR ENHANCING DISCOURSE: The
teacher of mathematics, in order to enhance
discourse, should encourage and accept the use
of•
•
•
•
Computers, calculators, and other technology
Concrete materials used as models
Pictures, diagrams, tables, and graphs
Metaphors, analogies, and stories
LEARNING ENVIRONMENT: The teacher of
mathematics should create a learning
environment that fosters the development of each
student’s mathematical power by-
• Providing structure and time necessary to explore
and grapple with significant ideas and problems
• Using the physical space and materials to facilitate
student learning
• Respecting and valuing students’ ideas, ways of
thinking, and mathematical dispositions
and by consistently expecting and
encouraging students to-
• Work independently and collaboratively to make
sense of mathematics
• Take intellectual risks by raising questions and
posing conjectures
ANALYSIS OF TEACHING AND LEARNING:
the teacher of mathematics should engage in
ongoing analysis of teaching and learning by-
• Observing, listening to, and gathering information
about students to assess what they are learning
• Examining effects of tasks, discourse, and learning
environment on students’ mathematical
knowledge, skills, and dispositions
In order to-
Ensure that every student is learning
Challenge and extend students’ thinking
Adapt or change activities while teaching
Report to students, parents, and administrators
Sources:
ƒ Professional Standards for Teaching Mathematics. Copyright 1991 by National Council of Teachers of Mathematics.
ƒ Principles and Standards for Teaching Mathematics. Copyright 2000 by National Council of Teachers of Mathematics.
ƒ Principles and Standards for Teaching Mathematics – Outreach CD, 2nd Edition. Copyright 2001 by National Council of
Teachers of Mathematics.
- C. Hopkinson, 3/28/04
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Everyday Math (EDM)
Everyday Mathematics is a research-based curriculum
developed by the University of Chicago School Mathematics
Project. UCSMP was founded in 1983 during a time of
growing consensus that our nation was failing to provide its
students with an adequate mathematical education. The goal
of this on-going project is to significantly improve the
mathematics curriculum and instruction for all school children
in the U.S. Based on their findings, the authors established
several basic principles that have guided the development of
Everyday Mathematics. These principles are:
ƒ Students acquire knowledge and skills, and develop an
understanding of mathematics from their own experience.
Mathematics is more meaningful when it is rooted in real
life contexts and situations, and when children are given
the opportunity to become actively involved in learning.
Teachers and other adults play a very important role in
providing children with rich and meaningful mathematical
experiences.
ƒ Children begin school with more mathematical
knowledge and intuition than previously believed. A K-6
curriculum should build on this intuitive and concrete
foundation, gradually helping children gain an
understanding of the abstract and symbolic.
ƒ Teachers, and their ability to provide excellent
instruction, are the key factors in the success of any
program. Previous efforts to reform mathematics
instruction failed because they did not adequately
consider the working lives of teachers.
What are some key features of Everyday Math?
ƒ Problem solving for everyday situations
ƒ Developing readiness through hands-on activities
ƒ Establishing links between past experiences and
explorations of new concepts
ƒ Sharing ideas through discussion
ƒ Cooperative learning
ƒ Practice through games
ƒ Ongoing review throughout the year
ƒ Daily routines
ƒ Ongoing assessment
ƒ Home and school partnership
Connected Math Project (CMP)
The Connected Mathematics Project (CMP) was funded by
the National Science Foundation between 1991 and 1997 to
develop a mathematics curriculum for grades 6, 7, and 8. The
result was Connected Mathematics, a complete mathematics
curriculum that helps students develop understanding of
important concepts, skills, procedures, and ways of thinking
and reasoning in number, geometry, measurement, algebra,
probability, and statistics. Listed here are some key features of
Connected Mathematics:
ƒ It is problem-centered. Important mathematical concepts
are embedded in engaging problems. Students develop
understanding and skill as they explore the problems
individually, in a group, or with the class.
ƒ It provides skills practice. The in-class problems and
homework questions give students practice with
important concepts, skills, and algorithms.
ƒ It is complete. The Connected Mathematics unitsmultiple units for each grade-form a complete middle
school curriculum that develops mathematical skills and
conceptual understanding across mathematical strands. In
addition, the program provides a complete assessment
package that includes quizzes, tests, and projects.
ƒ It is for teachers as well as students. The Connected
Mathematics materials were written so teachers can learn
from them too. The Teacher's Guides include extensive
notes regarding mathematics, pedagogy, and assessment.
ƒ It is research based. Each Connected Mathematics unit
has been field tested, evaluated, and revised over a threeto four-year period. Approximately 160 teachers and
45,000 students in diverse school settings across the
United States participated in the development of the
curriculum.
ƒ It is effective. Research results consistently show that
CMP students outperform other students on tests of
problem-solving ability, conceptual understanding, and
proportional reasoning. And CMP students do as well as,
or better than, other students on tests of basic skills.
Source: Connected Mathematics Project Website
(http://www.math.msu.edu/cmp)
When should I expect mastery of a skill?
ƒ A concept builds within a school year and within the K-5
experience
ƒ Concept is introduced and then revisited in a new way
asking children to apply concepts over time, deepening
their understanding
ƒ 5 times in 2 years
ƒ Proficiency levels – Beginning, Developing, Secure
What are the components of Everyday Math?
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
Journal
Mental Math and Reflexes
Math Boxes
Student Reference Book
Homelinks
Games
Checking Progress and Profiles of Progress
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What is Computational Fluency?
“Children should master the basic facts of arithmetic that are essential components of fluency with
paper-and-pencil and mental computation and with estimation…. It is important for children to
learn the sequence of steps - and the reasons for them - in the paper-and-pencil algorithms used
widely in our culture.”
Curriculum and Evaluation Standards for School Mathematics, 1989, p. 47
• Efficiency implies that the student does not get bogged down in many steps or lose track of
the logic of the strategy. An efficient strategy is one that the student can carry out easily,
keeping track of sub-problems and using intermediate results to solve the problem.
• Accuracy means getting a correct answer. It depends on several things, including
knowledge of basic number combinations and number relationships, checking for
reasonableness, and, in some cases, careful recording of work.
• Flexibility means that students understand the underlying properties of the strategies they
are using. It means using approaches that are appropriate for the particular problem. It
generally means that a students uses multiple approaches to compute.
Students who demonstrate computational fluency:
•
•
•
•
•
demonstrate flexibility in choosing computational methods;
understand and can explain these methods;
produce accurate answers efficiently;
can represent their thinking and work;
exhibit number sense.
Principles of computational fluency:
•
•
•
•
•
•
•
•
Computational fluency is an essential goal for school mathematics.
The methods a student uses to compute should be grounded in understanding.
Students should know the basic number combinations for addition and subtraction by the
end of grade 2 and for multiplication and division by the end of grade 4.
Students should be computing fluently with whole numbers by the end of grade 5 and with
fractions and decimals by the end of grade 8.
Students can achieve computational fluency using a variety of methods and should, in fact,
be comfortable with more than one approach.
Students should have opportunities to invent strategies for computing on the basis of their
knowledge of place value, number properties and the operations.
Students should investigate conventional algorithms for computing whole numbers.
Students should be encouraged to use computational methods that are appropriate for the
context and purpose, including mental computation, estimation, calculator, or paper-andpencil.
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Number & Operations (N)
NCTM standard N1: Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Grades PreK-2
Grades 3-5
Grades 6-8
a. Count with understanding and
g. Understand the place value structure of n. Work flexibly with fractions, decimals,
recognize “how many” in sets of objects.
the base-ten number system and be able to
and percents to solve problems.
represent and compare whole numbers and
decimals.
b. Use multiple models to develop initial
understandings of place value and the baseten number system.
h. Recognize equivalent representations
for the same number and generate them by
decomposing and composing numbers.
o. Compare and order fractions, decimals,
and percents efficiently and find their
approximate locations on a number line.
c. Develop understanding of the relative
position and magnitude of whole numbers
and of ordinal and cardinal numbers and
their connections.
i.
Develop understanding of fractions as
parts of unit wholes, as parts of a collection,
as locations on number lines, and as
divisions of whole numbers.
p. Develop meaning for percents greater
than 100 and less than 1.
d. Develop a sense of whole numbers and
represent them in flexible ways, including
relating, composing, and decomposing
numbers.
j.
Use models, benchmarks, and
equivalent forms to judge the size of
fractions.
q. Understand and use ratios and
proportions to represent quantitative
relationships.
e. Connect number words and numerals
to the quantities they represent, using
various physical models and
representations.
k. Recognize and generate equivalent
forms of commonly used fractions,
decimals, and percents.
r.
Develop an understanding of large
numbers and recognize and appropriately
use exponential, scientific, and calculator
notation.
f.
Understand and represent commonly
used fractions, such as 1/4, 1/3, and 1/2.
l. Explore numbers less than 0 by
extending the number line and through
familiar applications.
s.
Use factors, multiples, prime
factorization, and relatively prime numbers
to solve problems.
m. Describe classes of numbers according
to characteristics such as the nature of their
factors.
t.
Develop meaning for integers and
represent and compare quantities with them.
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Grades 9-12
u. Develop a deeper understanding of
very large and very small numbers and of
various representations of them.
v. Compare and contrast the properties of
numbers and numbers systems, including
the rational and real numbers, and
understand complex numbers as solutions to
quadratic equations that do not have real
solutions.
w. Understand vectors and matrices as
systems that have some of the properties of
the real-number system.
x. Use number-theory arguments to
justify relationships involving whole
numbers.
7
Number & Operations (N)
NCTM standard N2: Understand meanings of operations and how they relate to one another.
Grades PreK-2
Grades 3-5
Grades 6-8
a. Understand various meanings of
d. Understand various meanings of
h. Understand the meaning and effects of
addition and subtraction of whole numbers
multiplication and division.
arithmetic operations with fractions,
and the relationship between the two
decimals, and integers.
operations.
Grades 9-12
k. Judge the effects of such operations as
multiplication, division, and computing
powers and roots on the magnitudes of
quantities.
b. Understand the effects of adding and
subtracting whole numbers.
e. Understand the effects of multiplying
and dividing whole numbers.
i.
Use the associative and commutative
properties of addition and multiplication and
the distributive property of multiplication
over addition to simplify computations with
integers, fractions, and decimals.
l.
Develop an understanding of properties
of, and representations for, the addition and
multiplication of vectors and matrices.
c. Understand situations that entail
multiplication and division, such as equal
groupings of objects and sharing equally.
f. Identify and use relationships between
operations, such as division as the inverse of
multiplication, to solve problems.
j.
Understand and use the inverse
relationships of addition and subtraction,
multiplication and division, and squaring
and finding square roots to simplify
computations and solve problems.
m. Develop an understanding of
permutations and combinations as counting
techniques.
g. Understand and use properties of
operations, such as the distributivity of
multiplication over addition.
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Number & Operations (N)
NCTM standard N3: Compute fluently and make reasonable estimates.
Grades PreK-2
Grades 3-5
a. Develop and use strategies for wholed.
Develop fluency with basic number
number computations, with a focus on
combinations for multiplication and division
addition and subtraction.
and use these combinations to mentally
compute related problems, such as 30 x 50.
Grades 6-8
j.
Select appropriate methods and tools
for computing with fractions and decimals
from among mental computation,
estimation, calculators or computers, and
paper and pencil, depending on the
situation, and apply the selected methods.
Grades 9-12
n. Develop fluency in operations with real
numbers, vectors, and matrices, using
mental computation or pencil-and-paper
calculations for simple cases and technology
for more-complicated cases.
o. Judge the reasonableness of numerical
computations and their results.
b. Develop fluency with basic number
combinations for addition and subtraction.
e.
Develop fluency in adding,
subtracting, multiplying, and dividing whole
numbers.
k.
Develop and analyze algorithms for
computing with fractions, decimals, and
integers and develop fluency in their use.
c. Use a variety of methods and tools to
compute, including objects, mental
computation, estimation, paper and pencil,
and calculators.
f.
Develop and use strategies to estimate
the results of whole-number computations
and to judge the reasonableness of such
results.
l.
Develop and use strategies to estimate
the results of rational-number computations
and judge the reasonableness of the results.
g.
Develop and use strategies to estimate
computations involving fractions and
decimals in situations relevant to students’
experience.
m. Develop, analyze, and explain methods
for solving problems involving proportions,
such as scaling and finding equivalent
ratios.
h.
Use visual models, benchmarks, and
equivalent forms to add and subtract
commonly used fractions and decimals.
i.
Select appropriate methods and tools
for computing with whole numbers from
among mental computation, estimation,
calculators, and paper and pencil according
to the context and nature of the computation
and use the selected method or tool.
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Algebra (A)
NCTM standard A1: Understand patterns, relations, and functions.
Grades PreK-2
Grades 3-5
a. Sort, classify, and order objects by size, d. Describe, extend, and make
number, and other properties.
generalizations about geometric and
numeric patterns.
b. Recognize, describe, and extend
patterns such as sequences of sounds and
shapes or simple numeric patterns and
translate from one representation to another.
e. Represent and analyze patterns and
functions, using words, tables, and graphs.
c. Analyze how both repeating and
growing patterns are generated.
Grades 6-8
f. Represent, analyze, and generalize a
variety of patterns with tables, graphs,
words, and, when possible, symbolic rules.
Grades 9-12
i. Generalize patterns using explicitly
defined and recursively defined functions.
g. Relate and compare different forms of
representation for a relationship.
j.
Understand relations and functions and
select, convert flexibly among, and use
various representations for them.
h. Identify functions as linear or nonlinear
and contrast their properties from tables,
graphs, or equations.
k. Analyze functions of one variable by
investigating rates of change, intercepts,
zeros, asymptotes, and local and global
behavior.
l.
Understand and perform
transformations such as arithmetically
combining, composing, and inverting
commonly used functions, using technology
to perform such operations on morecomplicated symbolic expressions.
m. Understand and compare the properties
of classes of functions, including
exponential, polynomial, rational,
logarithmic, and periodic functions.
n. Interpret representations of functions of
two variables.
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Algebra (A)
NCTM standard A2: Represent and analyze mathematical situations and structures using algebraic symbols.
Grades PreK-2
Grades 3-5
Grades 6-8
a.
Illustrate general principles and
c.
Identify such properties as
f.
Develop an initial conceptual
properties of operations, such as
commutativity, associativity, and
understanding of different uses of variables.
commutativity, using specific numbers.
distributivity and use them to compute with
whole numbers.
b.
Use concrete, pictorial, and verbal
representations to develop an understanding
of invented and conventional symbolic
notations.
Grades 9-12
j.
Understand the meaning of
equivalent forms of expressions, equations,
inequalities, and relations.
d.
Represent the idea of a variable as an
unknown quantity using a letter or a symbol.
g.
Explore relationships between
symbolic expressions and graphs of lines,
paying particular attention to the meaning of
intercept and slope.
k.
Write equivalent forms of equations,
inequalities, and systems of equations and
solve them with fluency—mentally or with
paper and pencil in simple cases and using
technology in all cases.
e.
Express mathematical relationships
using equations.
h.
Use symbolic algebra to represent
situations and to solve problems, especially
those that involve linear relationships.
l.
Use symbolic algebra to represent
and explain mathematical relationships.
i.
Recognize and generate equivalent
forms for simple algebraic expressions and
solve linear equations.
m.
Use a variety of symbolic
representations, including recursive and
parametric equations, for functions and
relations.
n.
Judge the meaning, utility, and
reasonableness of the results of symbol
manipulations, including those carried out
by technology
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Algebra (A)
NCTM standard A3: Use mathematical models to represent and understand quantitative relationships.
Grades PreK-2
Grades 3-5
Grades 6-8
a.
Model situations that involve the
b.
Model problem situations with
c.
Model and solve contextualized
addition and subtraction of whole numbers,
objects and use representations such as
problems using various representations,
using objects, pictures, and symbols.
graphs, tables, and equations to draw
such as graphs, tables, and equations.
conclusions.
Grades 9-12
d.
Identify essential quantitative
relationships in a situation and determine
the class or classes of functions that might
model the relationships.
e.
Use symbolic expressions, including
iterative and recursive forms, to represent
relationships arising from various contexts.
f.
Draw reasonable conclusions about a
situation being modeled.
Algebra (A)
NCTM standard A4: Analyze change in various contexts.
Grades PreK-2
Grades 3-5
a.
Describe qualitative change, such as a c.
Investigate how a change in one
student’s growing taller.
variable relates to a change in a second
variable.
b.
Describe quantitative change, such as
a student’s growing two inches in one year.
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Grades 6-8
e.
Use graphs to analyze the nature of
changes in quantities in linear relationships.
Grades 9-12
f.
Approximate and interpret rates of
change from graphical and numerical data.
d.
Identify and describe situations with
constant or varying rates of change and
compare them.
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Geometry (G)
NCTM standard G1: Analyze characteristics and properties of 2- and 3-dimensional geometric shapes and develop mathematical arguments about geometric relationships.
Grades PreK-2
a.
Recognize, name, build, draw,
compare, and sort two- and threedimensional shapes.
Grades 3-5
d.
Identify, compare, and analyze
attributes of two- and three-dimensional
shapes and develop vocabulary to describe
the attributes.
Grades 6-8
i.
Precisely describe, classify, and
understand relationships among types of
two- and three-dimensional objects using
their defining properties.
Grades 9-12
l.
Analyze properties and determine
attributes of two- and three-dimensional
objects.
b.
Describe attributes and parts of twoand three-dimensional shapes.
e.
Classify two- and three-dimensional
shapes according to their properties and
develop definitions of classes of shapes such
as triangles and pyramids.
j.
Understand relationships among the
angles, side lengths, perimeters, areas, and
volumes of similar objects.
m.
Explore relationships (including
congruence and similarity) among classes of
two- and three-dimensional geometric
objects, make and test conjectures about
them, and solve problems involving them.
c.
Investigate and predict the results of
putting together and taking apart two- and
three-dimensional shapes.
f.
Investigate, describe, and reason
about the results of subdividing, combining,
and transforming shapes.
k.
Create and critique inductive and
deductive arguments concerning geometric
ideas and relationships, such as congruence,
similarity, and the Pythagorean relationship.
n.
Establish the validity of geometric
conjectures using deduction, prove
theorems, and critique arguments made by
others.
g.
Explore congruence and similarity.
o.
Use trigonometric relationships to
determine lengths and angle measures.
h.
Make and test conjectures about
geometric properties and relationships and
develop logical arguments to justify
conclusions.
Geometry (G
NCTM standard G2: Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
Grades PreK-2
Grades 3-5
Grades 6-8
Grades 9-12
a.
Describe, name, and interpret relative d.
Describe location and movement
g.
Use coordinate geometry to represent i.
Use Cartesian coordinates and other
positions in space and apply ideas about
using common language and geometric
and examine the properties of geometric
coordinate systems, such as navigational,
relative position.
vocabulary.
shapes.
polar, or spherical systems, to analyze
geometric situations.
b.
Describe, name, and interpret
direction and distance in navigating space
and apply ideas about direction and
distance.
e.
Make and use coordinate systems to
specify locations and to describe paths.
c.
Find and name locations with simple
relationships such as “near to” and in
coordinate systems such as maps.
f.
Find the distance between points
along horizontal and vertical lines of a
coordinate system.
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h.
Use coordinate geometry to examine
special geometric shapes, such as regular
polygons or those with pairs of parallel or
perpendicular sides.
j.
Investigate conjectures and solve
problems involving two- and threedimensional objects represented with
Cartesian coordinates.
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Geometry (G)
NCTM standard G3: Apply transformations and use symmetry to analyze mathematical situations.
Grades PreK-2
Grades 3-5
Grades 6-8
a.
Recognize and apply slides, flips, and c.
Predict and describe the results of
f.
Describe sizes, positions, and
turns.
sliding, flipping, and turning twoorientations of shapes under informal
dimensional shapes.
transformations such as flips, turns, slides,
and scaling.
b.
Recognize and create shapes that
have symmetry.
d.
Describe a motion or a series of
motions that will show that two shapes are
congruent.
g.
Examine the congruence, similarity,
and line or rotational symmetry of objects
using transformations.
Grades 9-12
h.
Understand and represent
translations, reflections, rotations, and
dilations of objects in the plane using
sketches, coordinates, vectors, function
notation, and matrices.
i.
Use various representations to help
understand the effects of simple
transformations and their compositions.
e.
Identify and describe line and
rotational symmetry in two- and threedimensional shapes and designs.
Geometry (G)
NCTM standard G4: Use visualization, spatial reasoning, and geometric modeling to solve problems.
Grades PreK-2
Grades 3-5
Grades 6-8
a.
Create mental images of geometric
e.
Build and draw geometric objects.
k.
Draw geometric objects with
shapes using spatial memory and spatial
specified properties, such as side lengths or
visualization.
angle measurements.
Grades 9-12
p.
Draw and construct representations
of two- and three-dimensional geometric
objects using a variety of tools.
b.
Recognize and represent shapes from
different perspectives.
f.
Create and describe mental images of
objects, patterns, and paths.
l.
Use two-dimensional representations
of three-dimensional objects to visualize
and solve problems such as those involving
surface area and volume.
q.
Visualize three-dimensional objects
from different perspectives and analyze their
cross sections.
c.
Relate ideas in geometry to ideas in
number and measurement.
g.
Identify and build a threedimensional object from two-dimensional
representations of that object.
m.
Use visual tools such as networks to
represent and solve problems.
r.
Use vertex-edge graphs to model and
solve problems.
d.
Recognize geometric shapes and
structures in the environment and specify
their location.
h.
Identify and build a two-dimensional
representation of a three-dimensional object.
n.
Use geometric models to represent
and explain numerical and algebraic
relationships.
s.
Use geometric models to gain
insights into, and answer questions in, other
areas of mathematics.
i.
Use geometric models to solve
problems in other areas of mathematics,
such as number and measurement.
o.
Recognize and apply geometric ideas
and relationships in areas outside the
mathematics classroom, such as art, science,
and everyday life.
t.
Use geometric ideas to solve
problems in, and gain insights into, other
disciplines and other areas of interest such
as art and architecture.
j.
Recognize geometric ideas and
relationships and apply them to other
disciplines and to problems that arise in the
classroom or in everyday life.
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Measurement (M)
NCTM standard M1: Understand measurable attributes of objects and the units, systems, and processes of measurement.
Grades PreK-2
Grades 3-5
Grades 6-8
a.
Recognize the attributes of length,
e.
Understand such attributes as length,
j.
Understand both metric and
volume, weight, area, and time.
area, weight, volume, and size of angle and
customary systems of measurement.
select the appropriate type of unit for
measuring each attribute.
b.
Compare and order objects according
to these attributes.
f.
Understand the need for measuring
with NCTM standard units and become
familiar with NCTM standard units in the
customary and metric systems.
k.
Understand relationships among units
and convert from one unit to another within
the same system.
c.
Understand how to measure using
non-NCTM standard and NCTM standard
units.
g.
Carry out simple unit conversions,
such as from centimeters to meters, within a
system of measurement.
l.
Understand, select, and use units of
appropriate size and type to measure angles,
perimeter, area, surface area, and volume.
d.
Select an appropriate unit and tool
for the attribute being measured.
h.
Understand that measurements are
approximations and understand how
differences in units affect precision.
Grades 9-12
m.
Make decisions about units and scales
that are appropriate for problem situations
involving measurement.
i.
Explore what happens to
measurements of a two-dimensional shape
such as its perimeter and area when the
shape is changed in some way.
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Measurement (M)
NCTM standard M2: Apply appropriate techniques, tools, and formulas to determine measurements.
Grades PreK-2
Grades 3-5
Grades 6-8
a.
Measure with multiple copies of units e.
Develop strategies for estimating the
j.
Use common benchmarks to select
of the same size, such as paper clips laid end perimeters, areas, and volumes of irregular
appropriate methods for estimating
to end.
shapes.
measurements.
Grades 9-12
p.
Analyze precision, accuracy, and
approximate error in measurement
situations.
b.
Use repetition of a single unit to
measure something larger than the unit, for
instance, measuring the length of a room
with a single meter stick.
f.
Select and apply appropriate NCTM
standard units and tools to measure length,
area, and volume, weight, time, temperature,
and the size of angles.
k.
Select and apply techniques and tools
to accurately find length, area, volume, and
angle measures to appropriate levels of
precision.
q.
Understand and use formulas for the
area, surface area, and volume of geometric
figures, including cones, spheres, and
cylinders.
c.
g.
Select and use benchmarks to
estimate measurements.
l.
Develop and use formulas to
determine the circumference of circles and
the area of triangles, parallelograms,
trapezoids, and circles and develop
strategies to find the area of more-complex
shapes.
m.
Develop strategies to determine the
surface area and volume of selected prisms,
pyramids, and cylinders.
r.
Apply informal concepts of
successive approximation, upper and lower
bounds, and limit in measurement
situations.
Use tools to measure.
d.
Develop common referents for
measures to make comparisons and
estimates.
h.
Develop, understand, and use
formulas to find the area of rectangles and
related triangles and parallelograms.
i.
Develop strategies to determine the
surface areas and volumes of rectangular
solids.
s.
Use unit analysis to check
measurement computations.
n.
Solve problems involving scale
factors, using ratio and proportion.
o.
Solve simple problems involving
rates and derived measurements for such
attributes as velocity and density.
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Data Analysis & Probability
NCTM standard D1: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
Grades PreK-2
Grades 3-5
Grades 6-8
Grades 9-12
a.
Pose questions and gather data about
d.
Design investigations to address a
h.
Formulate questions, design studies,
j.
Understand the differences among
themselves and their surroundings.
question and consider how data collection
and collect data about a characteristic shared various kinds of studies and which types of
methods affect the nature of the data set.
by two populations or different
inferences can legitimately be drawn from
characteristics within one population.
each.
b.
Sort and classify objects according to
their attributes and organize data about the
objects.
e.
Collect data using observations,
surveys, and experiments.
i.
Select, create, and use appropriate
graphical representations of data, including
histograms, box plots, and scatter plots.
k.
Know the characteristics of welldesigned studies, including the role of
randomization in surveys and experiments.
c.
Represent data using concrete
objects, pictures, and graphs.
f.
Represent data using tables and
graphs such as line plots, bar graphs, and
line graphs.
l.
Understand the meaning of
measurement data and categorical data, of
univariate and bivariate data, and of the
term variable.
g.
Recognize the differences in
representing categorical and numerical data.
m.
Understand histograms, parallel box
plots, and scatterplots and use them to
display data.
n.
Compute basic statistics and
understand the distinction between a
statistic and a parameter.
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Data Analysis & Probability (D)
NCTM standard D2: Select and use appropriate statistical methods to analyze data.
Grades PreK-2
Grades 3-5
a.
Describe parts of the data and the set
b.
Describe the shape and important
of data as a whole to determine what the
features of a set of data and compare related
data show.
data sets, with an emphasis on how the data
are distributed.
c.
Use measures of center, focusing on
the median, and understand what each does
and does not indicate about the data set.
d.
Compare different representations of
the same data and evaluate how well each
representation shows important aspects of
the data.
Grades 6-8
e.
Find, use, and interpret measures of
center and spread, including mean and
interquartile range.
Grades 9-12
g.
For univariate measurement data, be
able to display the distribution, describe its
shape, and select and calculate summary
statistics.
f.
Discuss and understand the
correspondence between data sets and their
graphical representations, especially
histograms, stem-and-leaf plots, box plots,
and scatter plots.
h.
For bivariate measurement data, be
able to display a scatterplot, describe its
shape, and determine regression
coefficients, regression equations, and
correlation coefficients using technological
tools.
i.
Display and discuss bivariate data
where at least one variable is categorical.
j.
Recognize how linear
transformations of univariate data affect
shape, center, and spread.
k.
Identify trends in bivariate data and
find functions that model the data or
transform the data so that they can be
modeled.
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Data Analysis & Probability (D)
NCTM standard D3: Develop and evaluate inferences and predictions that are based on data.
Grades PreK-2
a.
Discuss events related to students’
experiences as likely or unlikely.
Grades 3-5
b.
Propose and justify conclusions and
predictions that are based on data and design
studies to further investigate the conclusions
or predictions.
Grades 6-8
c.
Use observations about differences
between two or more samples to make
conjectures about the populations from
which the samples were taken.
Grades 9-12
f.
Use simulations to explore the
variability of sample statistics from a known
population and to construct sampling
distributions.
d.
Make conjectures about possible
relationships between two characteristics of
a sample on the basis of scatterplots of the
data and approximate lines of fit.
g.
Understand how sample statistics
reflect the values of population parameters
and use sampling distributions as the basis
for informal inference.
e.
Use conjectures to formulate new
questions and plan new studies to answer
them.
h.
Evaluate published reports that are
based on data by examining the design of
the study, the appropriateness of the data
analysis, and the validity of conclusions.
i.
Understand how basic statistical
techniques are used to monitor process
characteristics in the workplace.
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Data Analysis & Probability (D)
NCTM standard D4: Understand and apply basic concepts of probability.
Grades PreK-2
Grades 3-5
a.
Describe events as likely or unlikely
and discuss the degree of likelihood using
such words as certain, equally likely, and
impossible.
Grades 6-8
d.
Understand and use appropriate
terminology to describe complementary and
mutually exclusive events.
Grades 9-12
g.
Understand the concepts of sample
space and probability distribution and
construct sample spaces and distributions in
simple cases.
b.
Predict the probability of outcomes
of simple experiments and test the
predictions.
e.
Use proportionality and a basic
understanding of probability to make and
test conjectures about the results of
experiments and simulations.
h.
Use simulations to construct
empirical probability distributions.
c.
Understand that the measure of the
likelihood of an event can be represented by
a number from 0 to 1.
f.
Compute probabilities for simple
compound events, using such methods as
organized lists, tree diagrams, and area
models.
i.
Compute and interpret the expected
value of random variables in simple cases.
j.
Understand the concepts of
conditional probability and independent
events.
k.
Understand how to compute the
probability of a compound event.
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Pre-K-12 NCTM Process Standards
NCTM standard
Problem Solving (PRS)
Reasoning (RES)
Communication (COM)
Connections (CON)
Representation (REP)
CSSU Curriculum Frameworks – May 2004
Expectation
PRS1: Build new mathematical knowledge through problem solving
PRS2: Solve problems that arise in mathematics and in other contexts
PRS3: Apply and adapt a variety of appropriate strategies to solve problems
PRS4: Monitor and reflect on the process of mathematical problem solving
RES1: Recognize reasoning and proof as fundamental aspects of mathematics
RES2: Make and investigate mathematical conjectures
RES3: Develop and evaluate mathematical arguments and proofs
RES4: Select and use various types of reasoning and methods of proof
COM1: Organize and consolidate their mathematical thinking through communication
COM2: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
COM3: Analyze and evaluate the mathematical thinking and strategies of others
COM4: Use the language of mathematics to express mathematical ideas precisely
CON1: Recognize and use connections among mathematical ideas
CON2: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
CON3: Recognize and apply mathematics in contexts outside of mathematics
REP1: Create and use representations to organize, record, and communicate mathematical ideas
REP2: Select, apply, and translate among mathematical representations to solve problems
REP3: Use representations to model and interpret physical, social, and mathematical phenomena
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